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The spectral picture and the closure of the similarity orbit of strongly irreducible operators
Authors:Chun-Ian Jiang  Zong-yao Wang
Institution:(1) Department of Mathematics, Jilin University, 130023 Changchun, P. R. China;(2) Department of Mathematics, East China University of Science and Technology, 200237 Shanghai, P. R. China
Abstract:An operator on a complex, separable, infinite dimensional Hilbert space is strongly irreducible if it does not commute with any nontrivial idempotent. This article answers the following questions of D. A. Herrero: (i) Given an operatorT with connected spectrum, can we find a strongly irreducible operatorL such that they have same spectral picture? (ii) When we use a sequence of irreducible operators to approximateT, can the approximation be the “most economic”? i.e., does there exist a strongly irreducible operatorL such thatTS(L) ? (the closure of the similarity orbit ofL)? It is shown that the answer for the two questions is yes.
Keywords:Primary 47A15  Secondary 47A10
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